To determine the performance or behavior of a power system under unbalanced conditions it is essential to know sequence impedance and network of power system elements. This is a piece of fundamental knowledge for performing the unsymmetrical fault analysis in the power system. Here we will be discussing the sequence impedance and networks of various power system elements. 

Sequence Impedance and Network of Power System Elements: Synchronous Machine

For determining the sequence impedance and network of power system elements, let us consider an unloaded synchronous machine which can be a generator or motor. It is grounded through a reactor of impedance Zn. Ea, Eb, and Ec are the EMFs induce in the three phases of the machine respectively.

Sequence Impedance and Network of Power System Elements

Figure: 3 Phase Unloaded Synchronous Machine

An unsymmetrical fault on the machine terminals causes unbalanced current Ia, Ib, and Ic to flow in the respective phases. If this fault involves the ground then-current Inflows to neutral from the ground through reactor Zn. The current In is the phasor sum of Ia, Ib, and Ic.

These unbalanced currents can be resolved into symmetrical currents and respective positive, negative, and zero sequence networks of the machine can be formed.

Positive Sequence Impedance and Network of Synchronous Machine

Synchronous machines offer time-varying reactance Xd”(Sub-transient reactance), Xd‘(transient reactance) and Xd(synchronous reactance).

For a circuit where the sudden value of current under switching or fault condition is to be determined, the Sub-transient reactance of the machine is used. To determine the current after the few-cycle of switching or fault (3 or 4 cycles), transient reactance Xd‘ is used. For steady-state conditions, synchronous reactance Xd is used.

Positive sequence impedance along with the reactance of the machine makes up positive sequence sub transient, transient, or steady-state impedance of the machine respectively.

Given is the positive sequence network of synchronous machine

Sequence Impedance and Network of Power System Elements

Figure: Positive Sequence Network of a Synchronous Machine

Sequence Impedance and Network of Power System Elements

Figure: Positive Sequence Network of a Synchronous Machine

The positive sequence impedance of the synchronous machine is represented by a source emf on no load and the positive sequence impedance of the machine. The phasor sum of positive sequence currents of the three phases Ia1, Ib1, Ic1 are zero, so no current flows through neutral and Zn doesn’t appear in the network.

Negative Sequence Impedance and Network of Synchronous Machine

Negative sequence reactance X2 of the synchronous machine oscillates between the direct axis reactance Xd”and quadrature axis reactance Xq”. X2 value is usually taken average and given by

Figure: Negative Sequence Network of a Synchronous Machine

Figure: Negative Sequence Network of a Synchronous Machine

Zero Sequence Impedance and Network of Synchronous Machine

The current flowing in the neutral through reactor impedance Zn is the sum of the zero-sequence currents in all three phases. So, the voltage drop caused by this sum of zero sequence current is 3Ia0Zn. The voltage drop of the zero-sequence terminal is 3Ia0Zn + Ia0Zg0. The zero sequence impedance is given by Zg0.

The zero-sequence network of the transmission line is shown below.

Figure: Zero Sequence Network of a Synchronous Machine

Figure: Zero Sequence Network of a Synchronous Machine

Zero sequence voltage of terminal ‘a’ is

Sequence Impedance and Network of Power System Elements: Transmission Line

For a transmission line, positive and negative sequence impedances are equal. Zero sequence impedance includes the impedance of the return path through the ground is different from positive and negative sequence impedance. Zero sequence impedance of transmission line is much larger than the positive and negative sequence impedance.

Figure: Zero Sequence Network

Figure: Positive Sequence Network

Figure: Negative Sequence Network

For a fully transposed transmission line, there is no mutual coupling between the sequence networks.

Sequence Impedance and Network of Loads

A star-connected load with isolated neutral has no path for the flow of zero sequence current. It offers infinite impedance between the neutral and ground leaving behind the zero-sequence network open-circuited between the neutral of the star-connected load and the reference bus.

Figure: Star Connected Load With Isolated Neutral

Figure: Zero Sequence Network

When the neutral of the star-connected load is grounded through a reactor of impedance Zn then zero-sequence voltage drop will occur due to the flow of zero sequence current through Zn given by 3Ia0 Zn. This is the same as that of Ia0 flowing through 3Zn. So, an impedance of 3Zn is introduced between the neutral terminal and the reference bus of the zero-sequence network as shown.

Figure: Zero Sequence Network for Star Connected Load With Neutral Grounded Through Impedance Zn

For a balanced load, the positive, negative, and zero-sequence impedance of the load are equal.

Figure: Positive/Negative Sequence Network For Star Connected Load


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